GRADING AND SCREENING 229 proper slope be given to the inclined deck shown in Fig. 3 the particles 1 to 12 will discharge along the edge BC in the order of size an'd position shown in the figure. This effect is obtained as must be evident by the faster rolling of the larger pieces while the longitudinal advance given the particles by the deck-motion mechanism is practically equal regardless of size. Before the particles are discharged from the side BC practically all diameter axes have been in contact with the grading surface, they tend therefore to report at edge BC according to average diameter and consequently more nearly according to volume than least size as would be the case in screening. A cube and a rectangular parallelepiped of the same volume would be expected to report together at the discharge edge in volumetric grading but could be separated by screening.1 McKesson-Rice Screenless Sizer.—Two types of this machine are shown in Figs. 6 and 7. Figure 6 shows a machine for making three sizes of coal from the bottom compartments and one or two more from the end B if this is desired and arrangements are made to that purpose. It is claimed for this machine that it can make five sizes from pieces averaging 6 in. thick down to fine slack and will handle from 40 to 80 tons an hour. Figure 7 shows a type of machine where the longitudinal advance is given by an endless travelling belt. In order to cause the particles to travel down over the fine corrugations of this machine the belt is oscillated rapidly with a device placed below it. The principal reason for adopting this type of machine on fine material is due to the difficulty of making fine dust advance under a differential shake. It has disadvantages which will be touched upon later. Ilolling friction is the chief principle of operation of the machine but to make the grading more perfect and to spread the discharge line, thus reducing the overlap at 1 There is no satisfactory theory of rolling friction. With all surfaces the actual lifting of the rolling body over their projections and irregularities consumes the energy of the rolling body or the energy to be applied to it to cause it to roll. The difficulty of stating the problem lies in the irregularity of the opposition offered by the surface. Such opposition cannot be reduced to a mathematical equation. If the rolling opposition iw negligible, which it never is, a formula for velocity may be deduced which may delight mathcniaticianH but hna no practical bearing on actualities and leads to the result that a circular ring rolls down an inclined plane of length L so as to have a terminal velocity of ^/gh. Down a frictionloHH piano it would have a terminal velocity of \/2oh and would not roll. The first formula leadH to tho conchiHion that the velocity is not connected with the diameter which is contrary to fact. Referring to 1%. 4 it i.s evident that the forces acting on the circular disc R are the gravitational component W sin 0 and the foree /'' resisting downward motion and applied tangentially to the disc at the point where it Teats on the plane. Tho unbalanced force is consequently W sin Q — F and since \V W this force acts on a mauu • W ain 0 — F equals ar, where a is the angular acceleration of the disc 11 u w and r ita radius. But as must be evident F also equals ar. On transposing, ar becomes equal to ^ "J" , tho linear acceleration of the periphery of the disc. Substituting this value in the general expression for velocity V equals \/2Mt, where a is the acceleration and s the space passed, V equals'^-----2~~' But h cqualH L sin 0, consequently V equals \/(/h. The explanation given by Trautwine as to the nature of rolling friction and the factors to which it is proportional IH about aa aatiafuctory as can be obtained. His conclusions do not however give any measure of it for reasons which have already been stated. The coefficient of rolling friction has not been determined even experimentally with any degree of satisfaction. Referring to Fig. 5, R is as before the disc and the components of the weight acting along and at right angles to tho plane are W sin 0 and W cos 0 respectively. To overcome the obstruction at b there must bo the relation that W sin Ocb equals W cos Oab. This will be evident if acb is considered a bell crank hinged at 6, when to obtain a balance the moments of the two pulls must be equal. As the depth of the obstruction becomes less the leverage of W sin 0 approaches r the radius of the disc. At the same time the moment W cos 0 ab approaches zero. The ability of a disc to overcome rolling friction consequently increases with its. diameter Also the friction is independent of the weight and specific gravity.